Why is the Lebesgue-Stieltjes measure a measure?

Firstly, note that the measure defined here is a Radon measure (that is $\lambda(B)<\infty$ for any bounded borel set $B$). hence it is also $\sigma$-finite (Because $\mathbb{R}=\bigcup_{n\in\mathbb{Z}}(n,n+1]$). So if I can only show that the measure $\lambda$ is $\sigma$-additive on the semifield $\{(a,b]:-\infty\leq a\leq b\leq\infty\}$ (showing this is trivial), then it would be so over $\mathcal{B}$ by Caratheodory Extension Theorem.